As a matter of "national" policy, the Confederate constitution forbade the reopening of the international slave trade. Article I, Section 9, Clause 1 of the Confederate constitution:

"The importation of negroes of the African race from any foreign country other than the slaveholding States or Territories of the United States of America, is hereby forbidden; and Congress is required to pass such laws as shall effectually prevent the same."

However, would this have remained in place after the war was over? As Jon L. Wakelyn writes in a footnote in his collection Southern Pamphlets on Secession, November 1860 - April 1861, the re-opening of the international slave trade was a "divisive issue among Southern states". In fact, that's precisely why it is addressed in the Confederate constitution. In that same book, Wakelyn reproduces a pamphlet authored by Robert Hardy Smith, a North Carolina delegate to the Confederate Constitutional Convention, in which Smith explains why the slave trade issue was addressed. The pamphlet is actually the text of a speech Smith gave in Alabama, explaining the reasoning for the clause to an audience that may not have been fully on board with it:

"We have dissolved the late Union chiefly because of the negro quarrel. Now, is there any man who wishes to reproduce that strife among ourselves? and yet does not he, who wished the slave trade left for the action of Congress, see that he proposed to open a Pandora's box among us and to cause our political arena again to resound with this discussion. Had we left the question unsettled, we should, in my opinion, have sown broadcast the seeds of discord and death in our Constitution. I congratulate the country that the strife has been put to rest forever, and that American slavery is to stand before the world as it is, and on its own merits."

Before the Secession Crisis of 1860-61, as Smith alludes to, the issue had Southern supporters on both sides, and was largely divided along a geographical line. Supporters for re-opening the trade were almost entirely from the Lower South. Opponents were concentrated in the Upper South, although by and large, Southerners were opposed to its re-opening.

Among elected officials, perhaps the most vocal proponent of re-opening the trade was William Lowndes Yancey, a former U.S. House member from Alabama in the 1840s who continued to be active in the Southern Democratic party's leadership, later serving as a Confederate Senator for Alabama in 1861-62. In a speech at the 1857 Southern Commercial Convention held in Knoxville, he argued:

"If slavery is right per se, if it is right to raise slaves for sale, does it not appear that it is right to import them? Let us then wipe from our statute book this mark of Cain which our enemies have placed there. We want negroes cheap, and we want a sufficiency of them, so as to supply the cotton demand of the whole world."

Among other proponents were James B. Owen (church minister, political activist, and author of the pamphlet "The Right, Cause, and Necessity for Secession" who would go on to serve in the Provisional Confederate Congress, representing Florida), Leonidas Spratt (editor of the Charleston Mercury, one of the leading pro-secession newspapers before the Civil War), Edmund Ruffin (author, pro-slavery political activist, and wealthy Virginia plantation owner), and J.D.B. De Bow (South Carolina-born, New Orleans-resident publisher of the pro-slavery De Bow's Review periodical).

In his book The Fire-Eaters, Eric H. Walther summarizes the arguments for re-opening the slave trade in the section of the book about De Bow. De Bow gave a speech at the 1859 Southern Commercial Convention in Vicksburg, about which Walther writes:

If African slavery were "very right and very proper," De Bow wondered, how was it "immoral, irreligious, wicked, and inexpedient" to bring more Africans to the South? Repeating information supplied by Edmund Ruffin, De Bow pointed to the rise in slave prices and increased demand for their labor as reasons to renew the trade. Natural population increase, he said, could not keep up with demand, especially in the rapidly expanding Southwest. Besides, an influx of slaves would bring prices down and enable more whites to buy them and use them on less profitable land. Finally, De Bow agreed with Yancey that federal restriction of slavery in any form was an insulting "brand upon the institutions of the South." Although only a small minority of southerners wished the African slave trade reopened, De Bow convinced the delegates at Vicksburg to pass a resolution, by a vote of 44 to 19, urging the repeal of all state and federal prohibitions on the trade.

Among opponents from slave states who wrote against re-opening the slave trade were Thomas L. Clingman (former U.S. House member and, at the time of secession, U.S. Senator from North Carolina), William C. Rives (career politician from Virginia, serving in the U.S. House, U.S. Senate, and even as U.S. Ambassador to France before joining the Confederate Congress), and Robert Jefferson Breckinridge (politician, Presbyterian minister, slaveholder, and staunch Unionist from Kentucky who was also the uncle of the Southern Democrats' 1860 presidential candidate John C. Breckinridge). In Wakelyn's book, he reproduces pamphlets from all three, and the main points they make in their pre-war arguments against the re-opening of the trade include:

• It would decrease the sales price of the currently-enslaved people, thereby, reducing the wealth of slaveholders, possibly even risking the profitability of slavery.

• It would endanger the security of the white South, appealing to the omnipresent fear of white Southerners against a "servile insurrection".

• The current enslaved black American population was culturally and socially superior to newly-arrived Africans, which would diminish the gains that the white slaveholder class had introduced to these "savages". (Writes Robert Hardy Smith: "From these [400,000 imported Africans] have sprung about four millions of improved, civilized, hardy and happy laborers...American slavery is to stand before the world as it is, and on its own merits." Writes R.J. Breckinridge: "Do you want some millions more of African cannibals thrown amongst you broadcast throughout the whole slave States?")

• It would only benefit the Lower South "Cotton States" and/or only the plantation class, while it would actually hurt the average middle-class Southern slaveholder. It would put the Lower South plantation class in an even more superior political position than they already were in, to the detriment of the rest of the white South.

• It would introduce political conflict between the Lower South and Upper South, threatening the future of a Confederate country before it even began. Therefore, it was better to maintain the status quo.

So, what did a post-war future hold for the Confederacy? It's hard to say. Questions about a post-war Confederacy are generally difficult to answer since the Confederacy never got anywhere very close to considering post-war questions. There were Confederate politicians on both sides of many issues, including this one. Aside from the broadest strokes of the goal of an independent country where slavery was preserved with a thriving agriculture industry, there wasn't much unanimity on visions for a post-war Confederate future. On just about every political issue, you can find one set of Confederates advancing post-war visions leaning one way, and others advancing a completely opposing vision.

To put it in perspective, the U.S. itself was four years removed from considering a Constitutional Convention after the American Revolution ended. The 1787 Constitutional Convention was held six years after the last major battle of that war. And even then, it came with some severe political conflict to get the U.S. Constitution ratified. Rhode Island didn't ratify the U.S. Constitution until 1790, nearly nine years after the Battle of Yorktown. The whole period was marked by serious political differences, which produced Federalist and Anti-Federalist political camps, with very different visions for the future of the United States of America.

Similarly, there's no telling if all the Confederate states would have been willing to maintain the status quo of the Confederate constitution, or if the international slave trade was an issue that would have sprung up again in the post-war period. It delves into speculative history, rather than actual history. It does suggest one of the issues that Northerners as well as Southern Unionists often cited as why secession was always a bad idea as a resolution to political conflict: if secession can be used to resolve political issues, what's to stop further secessions in the future? They had argued that the Civil War could just as well have been immediately followed by a split between the Lower South and Upper South. The future of the international slave trade was an issue that could possibly have caused further disunity. But since the Confederacy never really got beyond the immediate goal of trying to win independence, we'll never know what the Confederacy would have planned, much less actually have done, in the post-war period.

Further reading:

The African Slave Trade, Reopen Or Suppress, 1850-1860 by Clara Margaret Moeschler (it's an old Master's theses, but it's free, and deals directly with this question)